Add the following rational expressions. $\dfrac{8k^3}{7k+2}+\dfrac{11}{3k}=$
Explanation: We can add two rational expressions whose denominators are equal by adding the numerators and keeping the denominator the same. [Does this fit with how we add rational numbers?] When the denominators are not the same, we must manipulate them so that they become the same. In other words, we must find a common denominator. Since the two denominators do not share any common factors, the common denominator is simply the product of these two denominators: $({7k+2})\cdot({3k})$. Let's manipulate the expressions to have that denominator: $\begin{aligned} &\phantom{=}\dfrac{8k^3}{{7k+2}}+\dfrac{11}{{3k}} \\\\ &=\dfrac{8k^3\cdot({3k})}{({7k+2})\cdot({3k})}+\dfrac{11\cdot({7k+2})}{({3k})\cdot({7k+2})} \end{aligned}$ [Why did we do that?] Now that both denominators are the same, let's add! $\begin{aligned} &\phantom{=}\dfrac{8k^3\cdot(3k)}{(7k+2)\cdot(3k)}+\dfrac{11\cdot(7k+2)}{(3k)\cdot(7k+2)} \\\\ &=\dfrac{8k^3\cdot(3k)+11\cdot(7k+2)}{(7k+2)(3k)} \\\\ &=\dfrac{24k^4+77k+22}{(7k+2)(3k)} \end{aligned}$ In conclusion, $\dfrac{8k^3}{7k+2}+\dfrac{11}{3k}=\dfrac{24k^4+77k+22}{(7k+2)(3k)}$